Question: $ D = \left[\begin{array}{rr}3 & -2 \\ 5 & 3\end{array}\right]$ $ A = \left[\begin{array}{rrr}-2 & 4 & 3 \\ -2 & 3 & 2\end{array}\right]$ What is $ D A$ ?
Answer: Because $ D$ has dimensions $(2\times2)$ and $ A$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ D A = \left[\begin{array}{rr}{3} & {-2} \\ {5} & {3}\end{array}\right] \left[\begin{array}{rrr}{-2} & \color{#DF0030}{4} & \color{#9D38BD}{3} \\ {-2} & \color{#DF0030}{3} & \color{#9D38BD}{2}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{-2}\cdot{-2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{-2}\cdot{-2} & ? & ? \\ {5}\cdot{-2}+{3}\cdot{-2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rrr}{3}\cdot{-2}+{-2}\cdot{-2} & {3}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{3} & ? \\ {5}\cdot{-2}+{3}\cdot{-2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{3}\cdot{-2}+{-2}\cdot{-2} & {3}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{3} & {3}\cdot\color{#9D38BD}{3}+{-2}\cdot\color{#9D38BD}{2} \\ {5}\cdot{-2}+{3}\cdot{-2} & {5}\cdot\color{#DF0030}{4}+{3}\cdot\color{#DF0030}{3} & {5}\cdot\color{#9D38BD}{3}+{3}\cdot\color{#9D38BD}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-2 & 6 & 5 \\ -16 & 29 & 21\end{array}\right] $